Title: An introduction to q-series
Speaker: Matt Boylan
Abstract: In this talk, I will use classical techniques to prove some basic, but important identities relating infinite products and infinite series. These identities are typically expressed in terms of a variable q; they are prototypes for what we call "q-series". This talk may therefore be viewed as an introduction to q-series. If time permits, I will mention applications to partition functions.
Title: q-series and partitions
Speaker: Matt Boylan
Abstract: In this talk, we will interpret some of the classical q-series identities presented last week using partitions. Time permitting, we will present other lesser-known examples.
Title: Proofs of q-series identities using Ferrer's diagrams
Speaker: Matt Boylan
Abstract: One can represent a partition of a positive integer n graphically using a Ferrer's diagram. In this talk, I will give classical examples of how Ferrer's diagrams have been used to prove q-series identities, including Franklin's proof of Euler's Pentagonal Number Theorem.
Title: Lattice points close to a curve and squarefree numbers in short intervals
Speaker: Ognian Trifonov
Abstract: This talk will be introduction to squarefree numbers in short intervals, lattice points close to a curve, divided differences and the connections between them.
Title: Lattice points close to a curve and squarefree numbers in short intervals, II
Speaker: Ognian Trifonov
Abstract: We will show how to use divided differences to estimate the number of lattice points close to a curve and will prove that there exists a constant c such that the interval (x , x + cx^{2/9}] contains a squarefree number for all x > 1.
Title: Missing digits in squarefree numbers and roots of reciprocal 0,1-polynomials
Speaker: Michael Filaseta
Abstract: I will talk on two different topics, both on results that are now several years old. The first is joint work with Sergei Konyagin showing that squarefree numbers exist with specified missing digits. The second is work by John Konvalina and Valentin Matache showing that 0,1-polynomials that are reciprocal necessarily have a root on the unit circle.
Title: Ferris Bueller's Day Off, The Breakfast Club and other things you really should know
Speaker: Michael Filaseta
Abstract: It seems like many young people have missed out on some good old (or not-so-old) classics. This is true whether or not we are talking about books, movies, technology or mathematics. We look at a few interesting theorems and proofs from the past that may have been overlooked by our youth and probably shouldn't have been. This includes a nice formula of W. H. Mills that generates only primes and a resolution by C. Pomerance to two conflicting conjectures.
Title: A variant of the large sieve
Speaker: Harsh Mehta
Abstract: We look at variants of the standard sieving problem and and talk about what is expected versus what is known and what kind of results under the established method will allow us to show what is expected. The problem at hand is to estimate upper bounds on the set A that remains after sieving out intervals of length (p-1)/2 mod p for every odd prime p less than N^{1/2}.
Title: Studying the size of ill distributed sets in dimension 2
Speaker: Harsh Mehta
Abstract: We study the sieving problem addressed last time in two dimensions (i.e., if the elements of a set S lie in at most alpha p residue classes mod p for all p > c, then how small is S?).
Title: Residue Completeness of Generalized Fibonacci Sequences mod m
Speaker: Jeremiah Southwick
Abstract: A generalized Fibonacci sequence is a sequence satisfying the Fibonacci relation. We call such a sequence complete mod m if all residues appear when the sequence is considered mod m. We will study this definition and will conclude by determining both necessary and sufficient conditions on the modulus m for a sequence to be complete mod m. Along the way we will highlight the importance of the Fibonacci numbers and the Lucas numbers, and will consider an invariant of generalized Fibonacci sequences that plays a key role in the results.
Title: Residue Completeness of Generalized Fibonacci Sequences mod m, Part II
Speaker: Jeremiah Southwick
Abstract: We continue a discussion of results about generalized Fibonacci sequences modulo m.